cs: O ani pe ch n e A c Delplace, Fluid Mech Open Acc 2016, 3:1 M c d e i s u s l DOI: 10.4172/2476-2296.1000125 F Fluid Mechanics: Open Access ISSN: 2476-2296 Research Article Open Access Reynolds Number and Spacetime Curvature Franck Delplace* ESI group scientific committee 100 Avenue de Suffren, Paris, France Abstract The reference length or scale length used in Reynolds number definition is of considerable importance. A quick review of Reynolds number definitions used in batch and continuous flow systems showed that this reference scale can be theoretical or conventional. Using curvature quantity coming from general relativity theory, we showed that Reynolds number could be seen as the ratio of two curvatures. This result could give interesting information for the design of high performances exchangers. Moreover, the use of curvature allowed establishing a relationship between momentum diffusivity and velocity gradient tensor. Applied to general relativity equation, we showed a strong link between gravity theory and hydrodynamics. Keywords: Reynolds; Navier-Stokes; Einstein; Curvature; Viscosity; y, z, t in Cartesian coordinates) deformed by stresses produced by the Reference length; Shear rate; Momentum diffusivity; Gravity; Fluid mechanical energy transferred to the liquid by the pumping or mixing dynamics device. Introduction The main objective of the present publication is then to revisit the Reynolds number definitions used for batch and continuous flow In fluid mechanics, the Reynolds number is certainly the most systems by use of the general relativity concept of curvature available important dimensionless quantity used in research and engineering. It for continuous spacetime medium. is defined as the ratio of momentum to viscous forces and it quantifies the relative importance of these two phenomena for given flow Reference Length in Reynolds Number Definitions conditions. When viscous forces are predominant, the flow regime is Fully established laminar flow of Newtonian fluids in pipes laminar and stream-lines are parallel to each other. Reynolds number values are low and for constant flow conditions in a duct of arbitrary Solving Navier-Stokes equation in that well known case gives cross section shape, stream-lines velocity varies from zero at the duct a parabolic velocity profile. For practical use, it is common to use a walls to a maximum value at the centre of the duct. Turbulent flow correlation between the Fanning friction factor f/2 and the Reynolds regime occurs at higher Reynolds number values when inertia forces number Re. This correlation often called friction law takes the following become predominant. In that conditions, stream-lines behaviour form: is more chaotic giving eddies, vortices and other flow instabilities. f 8 Between these well-defined flow regimes, there is an often called = (1) transition region delimited by critical Reynolds number values. As 2 Re shown by many authors (Leuliet [1], Delplace, [2] Delplace [3] critical with, Reynolds number values can vary a lot depending on ducts geometry. Changes of flow regime have a considerable importance for heat and f τ w ∆PD mass transfer industrial applications (heat exchangers, reactors.). = = (2) 2 ρρ u224 Lu From these well-established results, it is clear that geometry of flow systems is essential to explain and quantify flow phenomena. and, In the Reynolds number definition, geometry is described through a ρ uD "reference length" L sometimes called "scale length". Sometimes, L Re = (3) R R η can be determined analytically by solving Navier-Stokes equation, −1 this is the well-known case of fully developed laminar flow in a duct In these equations, τ w ()Pa is the wall shear stress, u () ms. is the ρ −3 ∆ of circular cross section shape where LR is the pipe diameter. But when mean fluid velocity, ()kg. m is the fluid density, P () Pa is the geometries are more complex, LR can be conventional. This is the pressure drop, Dm () is the pipe diameter L(m), is the pipe length and typical case of mixing tanks where L is the turbine diameter or a plate R η ()Pa. s is the dynamic viscosity. heat exchanger having complex flow geometry where LR is defined as twice the maximum gap between two plates (Leuliet [1], Delplace [2]). Recently, fluid mechanics concepts were introduced in theoretical *Corresponding author: Delplace F, ESI group scientific committee 100 Avenue de physics for general relativity (Padmanabhan [4], Delplace [5]) and Suffren, 75015 Paris, France, Tel: +33(0)615319558; E-mail: [email protected] quantum mechanics (Delplace [5]). From these results, it appears that Received May 30, 2016; Accepted June 10, 2016; Published June 18, 2016 Einstein general relativity equation is similar to Navier-Stokes equation and, if spacetime is considered as fluid; it could be possible to apply Citation: Delplace F (2016) Reynolds Number and Spacetime Curvature. Fluid Navier-Stokes equation at both astronomic and atomic scales. Mech Open Acc 3: 125. doi: 10.4172/2476-2296.1000125 Copyright: © 2016 Delplace F. This is an open-access article distributed under General relativity equation tells that spacetime geometry described the terms of the Creative Commons Attribution License, which permits unrestricted by its curvature (Einstein tensor) is proportional to stress energy use, distribution, and reproduction in any medium, provided the original author and tensor. In every flow system, one can consider we have a spacetime (x, source are credited. Fluid Mech Open Acc, an open access journal Volume 3 • Issue 1 • 1000125 ISSN: 2476-2296 Citation: Delplace F (2016) Reynolds Number and Spacetime Curvature. Fluid Mech Open Acc 3: 125. doi: 10.4172/2476-2296.1000125 Page 2 of 6 In that perfectly defined case, the reference length found for The case of cylindrical ducts of arbitrary cross section shape is often Reynolds number is the pipe diameter and we have LR=D. called two dimensional flows (2D). The case of three dimensional flows (3D) is much more complex because solving analytically the Navier- Fully established laminar flow of Newtonian fluids in ducts of Stokes equation is not possible. But the use of dimensionless numbers arbitrary cross section and particularly the Reynolds number gives powerful tools to engineers. In that case, Navier-Stokes equation reduces to the following well As a conclusion, for the laminar 2D flow in regular ducts of known Poisson partial differential equation: arbitrary cross section shape, we use LR=DH. This definition of scale ∆P length L can be considered as theoretical on the basis of equation (5). ∇=−2 uxy( ,) (4) R 2η L Fully established laminar flow of Newtonian fluids in complex u(x, y) is the scalar velocity field in the duct cross section, ΔP (Pa) is geometries the pressure drop L (m), is the pipe length and η (Pa. s) is the dynamic viscosity. As expressed above the case of 3D flows is much more complex than the case of 2D flows. 3D flows are encountered in continuous Even for simple geometries like rectangular or triangular ducts, flow systems in the case of complex shape channels like those built in analytical solutions of equation (4) are not simple. Shah et al. [6] gave compact heat exchangers. Because of their extensive use in chemical a complete review of these solutions for a wide variety of regular ducts. and food industries, plate heat exchangers are a good example of these For non regular ducts, only numerical approaches are available and are complex 3D flow geometries. For laminar flow conditions, plate heat always an intense field of research (Delplace [7] Sestak [8] Sochi [9]). exchangers were studied by many authors and reported in reference As reported by Midoux (1993), for a complex cross section shape, encyclopaedias (Cooper [11] Leuliet [1] Delplace [12]). a balance between pressure and friction forces allows determining f/2 For plate heat exchangers, all authors used as hydraulic diameter (equation 2) in a simple way: DH=2 e. e being the maximum gap between two plates. ξ values obtained f ∆PD experimentally varied a lot depending on plates geometry or design = H 2 (5) and on plate heat exchanger flow arrangement. Typical values varied in 2 4 Lu ρ the range 14 to more than 50. D (m) being the hydraulic diameter defined by: H Another important case of 3D flow is encountered in mixing 4 S systems or mixing tanks. In industrial reactors like fermentors, D = (6) H P turbines of various designs are used to create 3D flow patterns and obtain a maximum mixing efficiency. A huge amount of scientifical S (m2) is the cross section area and P(m) the wetted perimeter. work has been performed to predict power consumption of mixing This definition is of considerable importance because it allows systems. An accurate review and analyse of this work can be found in writing friction laws for ducts of arbitrary cross section in the following Delaplace [12]. Two dimensionless numbers defined below are used to general form in agreement with equation (1): characterise these complex flow systems: the power number Np and the f ξ rotational Reynolds number ReR. = (7) 2 Re P = N p 35 (10) As shown analytically and experimentally by many authors (Miller ρ ND [10] Leuliet [1] Delplace [3]), parameter ξ values depend only on ducts 2 ρ ND geometries. A good example is the case of rectangular ducts from ReR = (11) square duct to well known infinite parallel plates. Solving equation (4), η it is possible to calculate for each rectangle of cross section length 2b In these equations, P (W) is the power consumption of the turbine, and cross section width 2a the parameter ξ value as followed: N (rad.s-1) its rotational speed, ρ and η being respectively the fluid density and dynamic viscosity.